Automatic rational approximation and linearization of nonlinear eigenvalue problems

Abstract

We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

Algorithms for the Rational Approximation of Matrix-Valued Functions

“…Set-valued AAA (fast-AAA). The set-valued AAA algorithm presented in [23], and similarly the fast-AAA algorithm in [21], applies the standard AAA algorithm to each component of F using common weights and support points for all of them, thereby effectively producing a barycentric interpolant in the form (bary-A). Surrogate AAA.…”

“…6. Set j := j + 1 and go to step 2. https://github.com/nla-group/block aaa As it will become clearer from the discussions in section 4, the block-AAA algorithm is different from the set-valued AAA [23] and the fast-AAA [21] algorithms: the entries of a block-AAA approximant R d do not share a common scalar denominator of degree d. Indeed, a block-AAA approximant R d of order d can have a larger number of up to dm poles; i.e., its McMillan degree can be as high as dm. For this reason we refer to the order of a block-AAA approximant instead of its degree.…”

“…, d, as well as the greedy search in Step 2 which requires the evaluation of R j - 1 (z) at all remaining sampling points in \Lambda (j) to determine the next interpolation point z j . We have tried to speed up Step 5 using the updating trick of the SVD decomposition described in [23] but noticed some numerical instabilities for sufficiently large degrees. We have therefore not used any SVD updating strategy for the block-AAA results reported here.…”

Data-Driven Modeling of Linear Dynamical Systems with Quadratic Output in the AAA Framework

“…For further details, we refer the reader to the original source [32]. The AAA algorithm proved very flexible and effective, and has been employed in various applications such as rational approximation over disconnected domains [32], solving nonlinear eigenvalue problems [28], modeling of parametrized dynamics [13], and approximation of matrix-valued functions [20].…”

“…We note that state-space realizations for rational functions are unique up to a similarity transformation. For other equivalent state-space representations of a barycentric form, we refer the reader to, e.g., [5,28].…”